Probability theory does NOT regard imprecision to be equated
with randomness. That is an abuse of probability theory
admittedly indulged in all the time by Bayesians. But the
theory itself cannot be so implicated, and when the
Bayesians confound imprecision and randomness, they do it by
drawing *analogy* between belief uncertainty (including
specifically about that which is merely imprecise), and
randomness, using the notion of the betting paradigm to
explicate belief uncertainty. Analogy is not equation,
however, and analogies which only half fit often get us into
conceptual and other trouble. That was the first point I
tried to make, and I was rather crptic because I've made it
before, at perhaps too great length in various previous
posts.
The second point I wished to make was that when applied in
the frequentist set-up, probability theory *entails* a
*dual*, *different* notion of uncertainty as embodied in the
likelihood function. That makes probability more basic than
likelihood, in some sense, but not "omnipotent", whatever
that may be taken to mean. That latter, dual, notion of
uncertainty, the one embodied in a likelihood
function--which is in general a *point* funtion ranging over
*parameter* space, emphasizing the difference with a
probability function which is conceptually a *set* function
over *sample* space--is of the same sort, qualitatively, as
the membership function of a fuzzy set, which O'Sullivan
characterized, and I agree, as having to do with imprecision
rather than randomness. Thus, one may conceive of a random
sample as providing a fuzzy "measurement/description" of the
parameter of the probability model from which the random
sample is drawn, as embodied in the likelihood function.
The larger the random sample, the more precise the
"measurement", which, though, remains conceptually fuzzy.
This is akin to description/ measurement. A linguistic
measurement, eg. "tall" is qualitatively not different from
a numeric measurement, eg. 72 (+/- 0.5) inches, the
difference between the two being a matter of precision
appropriate to the measuring "instrument" being used--they
are both, however, ultiamately fuzzy, although in the one
case the fuzziness affects the first significant figure,
while being limited to only the second, in the other case.
Likewise, the "measurement" of a model parameter with a
random sample varies in precision directly with the size of
the random sample used--if you take a large enough sample,
your model parameter may be specified to as many significant
figures as you choose.
I am analogizing here, but unlike the Bayesians, who use
probability to characterize what are in fact likelihoods,
the analogy I am drawing is exact, which becomes apparent
once the grade of membership in a fuzzy set is identified
with the *probability* of word usage, of the fuzzy term in
question to describe/measure points from a relevant universe
of discourse. Such a collection of probabilities maps out
not a probability *set* function, but a likelihood or
membership *point* function over the space of hypotheses,
here identified with the universe of discourse to which the
term is denotatively linked. The confounding of which the
Bayesians are guilty is to combine probabilistic prior
("belief" probability by analogy only) with qualitatively
different likelihood (the data's way of saying, fuzzily,
what the model parameter is) to obtain a combined assessment
of prior data plus present data, taken together, to say,
still fuzzily--uncertainty of imprecision--what might be the
value of some model parameter of interest.
I could go on, but the broad outlines of what I am asserting
should be clear. As to your example, I heartily concur with
the interval analysis presented, and would add that a
possibilistic analysis deriving from fuzzy-as-likelihood
notions also would yield directly (natively) the result
demanded by your (and mine) intuition. As to the
difficulties of risk analysis, I am not against the
intrusion of subjectivist notions; I think, however, that
where priors are warranted, they should take the form,
qualitatively, of likelihood, which, as it happens, or at
least I assert, is a form of fuzzy, and vice versa. Much
conceptual clarification--to nagging foundational issues of
statistical inference on the one hand, and to equally
nagging foundational (interpretational) issues of fuzzy
logic on the other--follows from this realization.
I attach your entire post below as other readers would need
it to make sense of the exchange.
Regards,
S. F. Thomas
AB Risk Group (risk@life.bio.sunysb.edu) wrote:
: I've read many of your comments on this controversy and found them to be
: cogent and reasonable. However, I'm afraid I don't quite understand your
: response to O'Sullivan. The parameter for the (stationary) Bernoulli
: distribution that explains the frequency with which we get heads is a
: single number. I don't understand how your example in any way disabuses
: us of the idea that fuzzy methods and probability theory have different
: domains.
:
: Moreover, it seems clear to me at least, that classical probability theory
: is not the omnipotent calculus it is often portrayed to be. For instance,
: in risk analysis, where incomplete knowledge is the rule, one generally
: cannot get any answer at all without resorting to a subjectivist
: interpretation of probability, maximum entropy criteria, empirically
: unjustified assumptions of independence, or a combination of these and
: other strategies that are divorced still further from what can be
: justified by appeal to empirical information.
:
: Here is a simple example. Suppose I tell you that A and B are numerical
: inputs about whose values I am uncertain. I only know that the value or
: values of A must be between 0.3 and 0.5, and the value(s) of B must be
: between 0.2 and 0.4. What can be said about the sum A+B?
:
: Because I know nothing about A or B except their ranges, many Bayesians
: and most maximum entropy heads will use uniform distributions to model the
: two inputs and (maybe) some assumption of independence to compute a
: triangular distribution (0.5, 0.7, 0.9). This answer surely confounds
: ignorance (imprecision) with randomness (variability). Of course, some
: analysts wisely refuse to answer the question and produce no answer at
: all, although this is hardly a tenable strategy in general.
:
: As interval probability or robust Bayesian analysis or simple interval
: analysis reveals, the answer is the interval [0.5, 0.9]. In my view, this
: is the only correct answer from the point of view of a risk analysis who
: cannot use subjectivist interpretations but must rely on a pure
: frequentist interpretation. The interval corresponds to the *set* of
: probability distributions whose supports are on [0.5, 0.9], but not to any
: particular distribution from that set. It is the answer that is produced
: natively by fuzzy arithmetic (which generalizes interval analysis) but
: probability theory actually requires a rather sophisticated analysis to
: arrive at the correct conclusion.
:
: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
: Scott Ferson Telephone 516-751-4350
: Applied Biomathematics Facsimile 516-751-3435
: 100 North Country Road Email risk@life.bio.sunysb.edu
: Setauket, New York 11733 USA Web http://gramercy.ios.com/~ramas
: -----------------------------------------------------------------------
:
: On Fri, 24 May 1996, S. F. Thomas wrote:
:
: > Barry O'Sullivan (osullb@cs.ucc.ie) wrote:
: >
: >
: > : I think the easiest way to reconcile the differences between fuzzy
: > : membership and probability is as follows. Probability is based on
: > : randomness. Probability theory regards imprecision to be equated with
: > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
: > : randomness.
: > ^^^^^^^^^^
: > This is a common misconception. It is not so.
: >
: > : On the otherhand, fuzziness is the type of imprecision
: > : encountered when there does not exist a sharp transition between \
: > : membership and non-membership.
: >
: > Yes. Now consider the set of Bernoulli parameters which explain
: > the outcome on a single toss of say a thumb-tack for which we
: > wish to estimate the probability of it landing top down on a
: > single toss. Is there a sharp transition from membership to
: > non-membership for such a set? If you agree the answer is no,
: > think about what that implies for the relationship between
: > probability and fuzzy.
: >
: > : Kindest regards,
: > : Barry.
: >
: > Regards,
: > S. F. Thomas
: >
: >